Heat 2 tablespoons olive oil in a large skillet over medium-high heat. Add the onions, 1 teaspoon sugar, and ½ teaspoon salt and cook, stirring occasionally, for 15 minutes, or until the onions begin to brown.

Place 2 cups warm tap water in a large bowl and sprinkle with the yeast. Let stand 4 to 5 minutes, until it begins to foam. Whisk in the eggs, 4 tablespoons butter, the remaining ¼ cup sugar, and ½ teaspoon salt. Add the flour and ¾ of the onion and mix until the dough becomes sticky. Transfer the dough to a large bowl and brush the top with the remaining 1-tablespoon oil.

Cover the bowl with plastic wrap and let the dough rise 1 hour in a warm place until it doubles in bulk.

Butter a 9 x 13-inch baking pan.

Punch the dough down. With well-floured hands, form the dough into 16 balls and place in the prepared pan, spacing evenly. Sprinkle the rolls with the remaining onions. Cover the pan loosely with plastic wrap and let the dough rise in a warm place another 30 to 40 minutes, until it has doubled again.

Heat the oven to 400 degrees F.

Remove the plastic wrap and brush the dough with the remaining 2 tablespoons butter. Bake the rolls for 18 to 22 minutes, or until the tops are golden. (Place a foil tent over the rolls if the tops seem to be browning too quickly.) Let the rolls cool in the pan for 5 minutes before serving.